U.S. patent number 7,489,828 [Application Number 10/955,650] was granted by the patent office on 2009-02-10 for methods, system, and program product for the detection and correction of spherical aberration.
This patent grant is currently assigned to Media Cybernetics, Inc.. Invention is credited to Abu-Tarif Asad, Timothy J. Holmes.
United States Patent |
7,489,828 |
Asad , et al. |
February 10, 2009 |
Methods, system, and program product for the detection and
correction of spherical aberration
Abstract
The present invention provides methods, a system, and a program
product for the rapid detection and correction of spherical
aberration in microscopy systems. More specifically, the present
invention empirically derives a pupil function, adaptively corrects
PSF parameters, and automatically detects the coefficient for
spherical aberration. A first aspect of the invention provides a
method for detecting and estimating spherical aberration in an
acquired image obtained using an optical system, comprising the
steps of deconvolving an image using each of a plurality of point
spread functions, wherein each point spread function has a
different spherical aberration value, calculating an image energy
for each deconvolved image, and choosing as a spherical aberration
coefficient the spherical aberration value corresponding to the
deconvolved image having the lowest image energy, wherein a
spherical aberration coefficient other than 0 indicates the
presence of spherical aberration in the acquired image and its
distance from 0 is an estimation of the degree and direction of
spherical aberration.
Inventors: |
Asad; Abu-Tarif (Troy, NY),
Holmes; Timothy J. (East Greenbush, NY) |
Assignee: |
Media Cybernetics, Inc.
(Bethesda, MD)
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Family
ID: |
34526519 |
Appl.
No.: |
10/955,650 |
Filed: |
September 30, 2004 |
Prior Publication Data
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Document
Identifier |
Publication Date |
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US 20050083517 A1 |
Apr 21, 2005 |
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Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
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60508735 |
Oct 3, 2003 |
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Current U.S.
Class: |
382/254; 356/124;
382/255; 382/279 |
Current CPC
Class: |
G02B
21/365 (20130101); G02B 27/0025 (20130101) |
Current International
Class: |
G06K
9/40 (20060101); G06K 9/64 (20060101); G01B
9/00 (20060101) |
Field of
Search: |
;382/254 ;356/124 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
Primot et al., "Deconvolution from wave-front sensing: a new
technique for compensating turbulence-degraded images", Journal
Optical Society of America A, vol. 7, No. 9, Sep. 1990, pp.
1598-1608. cited by examiner .
Kabius, B. et al., "First Application of a Spherical-Aberration
Corrected Transmission Electron Microscope in Materials Science,"
Journal of Electron Microscopy, vol. 51, Supplement, 2002, pp.
S51-S58. cited by other .
O'Connor, N. J. et al., "Fluorescent Infrared Scanning-Laser
Ophthalmoscope for Three-Dimensional Visualization: Automatic
Random-Eye-Motion Correction and Deconvolution," Applied Optics,
vol. 37, No. 11, Apr. 10, 1998, pp. 2021-2033. cited by other .
Holmes, T. J., "Maximum-Likelihood Image Restoration Adapted for
Noncoherent Optical Imaging," Journal of the Optical Society of
America A, vol. 5, No. 5, May 1988, pp. 666-673. cited by other
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Gibson, S. F. et al., "Diffraction by a Circular Aperture as a
Model for Three-Dimensional Optical Microscopy," Journal of the
Optical Society of America A, vol. 6, No. 9, Sep. 1989, pp.
1357-1367. cited by other .
Holmes, T. J. et. al., "Acceleration of Maximum-Likelihood Image
Restoration for Fluorescence Microscopy and Other Noncoherent
Imagery," Journal of the Optical Society of America A, vol. 8, No.
6, Jun. 1991, pp. 893-907. cited by other .
Dellby, N. et al., "Progress in Aberration-Corrected Scanning
Transmission Electron Microscopy," Journal of Electron Microscopy,
vol. 50, No. 3, 2001, pp. 177-185. cited by other .
Krivanek, O.L. et al., "Towards sub-.ANG. Electron Beams,"
Ultramicroscopy, vol. 78, pp. 1-11, 1999. cited by other .
Markham, J. et al., "Parametric Blind Deconvolution: A Robust
Method for the Simultaneous Estimation of Image and Blur," J. Opt.
Soc. Am. A., vol. 16, No. 10, Oct. 1999, pp. 2377-2391. cited by
other .
Holmes, T. J., "Blind Deconvolution of Quantum-Limited Incoherent
Imagery: Maximum-Likelihood Approach," Journal of the Optical
Society of America A, vol. 9, No. 7, Jul. 1992, pp. 1052-1061.
cited by other .
Hopkins, H. H., "The Frequency Response of a Defocused Optical
System," Proc. R. Soc. A., 1955, pp. 91-103. cited by other .
Holmes, T. J., "Expectation-Maximization Restoration of
Band-Limited, Truncated Point-Process Intensities with Application
in Microscopy," Journal of the Optical Society of America A, 1989,
pp. 1006-1014. cited by other.
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Primary Examiner: Wu; Jingge
Assistant Examiner: Torres; Jose M
Attorney, Agent or Firm: Caesar, Rivise, Bernstein, Cohen
& Pokotilow, Ltd.
Parent Case Text
CROSS REFERENCE TO RELATED APPLICATIONS
The current application claims the benefit of co-pending U.S.
Provisional Application No. 60/508,735, filed Oct. 3, 2003, which
is hereby incorporated herein by reference.
Claims
What is claimed is:
1. A method for detecting and estimating spherical aberration in an
image obtained using an optical system, comprising the steps of:
deconvolving an acquired image using each of a plurality of point
spread functions, wherein each point spread function has a
different spherical aberration value, and wherein the spherical
aberration values are chosen from a range between -10 and 10;
calculating an image energy for each deconvolved image; and
choosing as a spherical aberration coefficient the spherical
aberration value corresponding to the deconvolved image having the
lowest image energy, wherein a spherical aberration coefficient
other than zero indicates the presence of spherical aberration in
the acquired image and its difference from zero is an estimation of
the degree and direction of spherical aberration.
2. The method of claim 1, wherein the optical system is a
microscopy system.
3. The method of claim 1, wherein the point spread function is
calculated from a pupil function formula.
4. The method of claim 3, wherein the pupil function formula is
.function.eI.function..times..times..rho..times..pi..times..times..times.-
.times..rho..rho..ltoreq..rho.> ##EQU00013## where k is a wave
number, n is a refraction index, A is an empirically derived value,
z is a distance from an in-focus to an out-of-focus plane, and S is
a spherical aberration value.
5. The method of claim 4, wherein the empirically derived value is
obtained by fitting a polynomial to data obtained by the steps of:
defining a range of hourglass angles; and calculating a pupil
function for each hourglass angle defined, wherein a value for A is
obtained that will yield the hourglass angle.
6. The method of claim 5, wherein the empirically derived
polynomial is a third order polynomial having the form
A.sub.empirical=c.sub.3.alpha..sup.3+c.sub.2.alpha..sup.2+C.sub.1.alpha.+-
c.sub.0, where .alpha. is the hourglass angle.
7. A method for detecting and estimating spherical aberration in an
image obtained using an optical system, comprising the steps of:
generating a plurality of spherically aberrated point spread
functions for an acquired image, wherein each PSF has a different
spherical aberration value; calculating an XZ sum-of-intensities
projection for each PSF; choosing an XZ and a YZ cross section of
the image corresponding to a maximum XZ intensity; calculating a
correlation coefficient using the PSF XZ projection and the XZ and
YZ cross-sections for each PSF; and choosing as a spherical
aberration coefficient the spherical aberration value associated
with the PSF corresponding to the highest correlation coefficient,
wherein a spherical aberration coefficient other than 0 indicates
the presence of spherical aberration in the acquired image and its
difference from 0 is an estimation of the degree and direction of
spherical aberration.
8. The method of claim 7, wherein the optical system is a
microscopy system.
9. The method of claim 7, wherein the PSF is calculated from a
pupil function formula.
10. The method of claim 9, wherein the pupil function formula is
.function.eI.function..times..times..rho..times..pi..times..times..times.-
.times..rho..rho..ltoreq..rho.> ##EQU00014## where k is a wave
number, n is a refraction index, A is an empirically derived value,
z is a distance from an in-focus to an out-of-focus plane, and S is
a spherical aberration value.
11. The method of claim 7, wherein values other than the spherical
aberration value are the same in each of the plurality of PSFs.
12. The method of claim 7, wherein the correlation coefficients are
calculated using the equation .times..times..times..times..times.
##EQU00015## where u={u.sub.i|i=1,2 . . . N} is the PSF X
projection and v={v.sub.i|i=1,2, . . . N} is one of the XZ data
cross-section and the YZ data cross-section.
13. A method for empirically deriving a point spread function
comprising the steps of: defining a range of hourglass angles;
calculating a pupil function for each hourglass angle using the
equation
.function.eI.function..times..times..rho..times..pi..times..times..times.-
.times..rho..rho..ltoreq..rho.> ##EQU00016## where k is a wave
number, n is a refraction index, z is a distance from an in-focus
to an out-of-focus plane, S is a spherical aberration value, and A
is a value that will yield the hourglass angle; fitting each value
for A to a polynomial equation; and deriving a point spread
function from the pupil function.
14. The method of claim 13, wherein the polynomial equation is
A.sub.empirical=c.sub.3.alpha..sub.3+c.sub.2.alpha..sup.2+C.sub.1.alpha.+-
c.sub.0, where .alpha. is the hourglass angle.
15. A system for detecting and correcting spherical aberration in
an image obtained using an optical system, comprising: a system for
detecting and estimating spherical aberration; a system for
empirically deriving a pupil function, wherein the system for
empirically deriving a pupil function includes a method comprising
the steps of: defining a range of hourglass angles; calculating a
pupil function for each hourglass angle defined using the equation
.function.eI.function..times..times..rho..times..pi..times..times..times.-
.times..rho..rho..ltoreq..rho.> ##EQU00017## where k is a wave
number, n is a refraction index, z is a distance from an in-focus
to an out-of-focus plane, S is a spherical aberration value, and A
is a value that will yield the hourglass angle; fitting each value
for A to a polynomial equation; substituting the polynomial
equation for the A factor in the pupil function equation; and using
the substituted pupil function equation to calculate a pupil
function; a system for deriving a point spread function from the
pupil function; and a system for deconvolving an imaged object from
an acquired image.
16. The system of claim 15, wherein the system for detecting and
estimating spherical aberration utilizes at least one of a
correlation-based method and an energy minimization method.
17. The system of claim 15, wherein the system for deconvolving an
imaged object from an acquired image includes at least one of a
blind deconvolution algorithm, a non-blind deconvolution algorithm,
and an inverse filter algorithm.
18. The system of claim 15, further comprising a system for
inputting acquired image data.
19. The system of claim 15, further comprising a system for
outputting processed data.
20. The system of claim 19, wherein the system for outputting
processed data includes at least one of a printer, a display
apparatus, and an electronic storage apparatus.
Description
BACKGROUND OF THE INVENTION
(1) Technical Field
The present invention relates generally to optical systems and more
specifically to methods, systems, and program products for the
detection and correction of spherical aberration in microscopy
systems.
(2) Related Art
Spherical aberration is a common problem in microscopy systems. It
is the result of differences in the focal points of light rays
based on their differing distances from the center of a lens, i.e.,
the optic axis of the lens. In the apparatus 100 shown in FIG. 1,
light rays 120 passing through a lens 110 at points further from
its center have a focal point 130 closer to lens 110 than the focal
point 132 formed by light rays 122 entering lens 110 at points
closer to its center. Often, this is caused, entirely or in part,
by a difference in the refractive indices of the lens immersion
medium and the embedding medium of the sample. While light rays
entering the lens at points further from its center are shown as
having focal points closer to the lens, such rays could also have
focal points further from the lens than do rays entering the lens
at points closer to its center. The effects of spherical aberration
include reductions in both the resolving power of the lens and the
signal-to-noise ratio in collected data. Spherical aberration
increases as one focuses further into a sample.
Spherical aberration is more problematic in spherical lenses.
Lenses having parabolic surfaces have been shown to reduce or
eliminate the effect of spherical aberration, but are not often
used, due to their great expense. Other methods for correcting
spherical aberration exist, including, for example, the use of a
correction collar on the lens, use of a lens immersion medium with
a refractive index matching that of the embedding medium, and
altering the thickness of the coverslip. The difficulty with such
methods, aside from the additional time and expense they require,
is that spherical aberration often is not detected until after
image data have been collected. Accordingly, the correction of
spherical aberration frequently occurs after data collection.
Traditionally, such correction has relied on the use of
deconvolution algorithms, which enable manipulation of an acquired
image to recover a more accurate representation of the imaged
object. Such algorithms generally follow the equation:
.mu..sub.n(x,y,z)=.chi..sub.n(x,y,z)*h(x,y,z)+b(x,y,z)+N(x,y,z)
(Eq. 1) where .mu..sub.n(x,y,z).ident.acquired image;
.chi..sub.n(x,y,z).ident.imaged object;
h(x,y,z).ident.point spread function (PSF);
b(x,y,z).ident.background level (primarily due to dark
current);
N(x,y,z).ident.random noise;
*.ident.convolution operator;
x,y.ident.in-plane spatial variables; and
z.ident.axial spatial variable.
Background level can be measured through a calibration protocol
described by Holmes et al. ("Light Microscopic Images Reconstructed
by Maximum Likelihood Deconvolution" in HANDBOOK OF BIOLOGICAL
CONFOCAL MICROSCOPY, 2d ed. (1995)) and random noise can be
statistically modeled. Thus, use of a deconvolution algorithm
essentially involves the recovery of an imaged object
(.chi..sub.n), given an acquired image (.mu..sub.n) and a
calculated PSF (h).
Deconvolution algorithms generally follow one of four schemes. The
first, described by Hopkins (The frequency response of a defocused
optical system, Proc. R. Soc. A., 91-103 (1955)), constructs a PSF
based on diffraction theory, using a generated pupil function. The
second, described by Holmes (Maximum-likelihood image restoration
adapted for noncoherent optical imaging, J. Opt. Soc. Am,
6:1006-1014 (1989)), assumes a measured PSF. The third, also
described by Holmes (Blind deconvolution of quantum-limited
incoherent imagery, J. Opt. Soc. Am., 9:1052-1061 (1992)),
concurrently estimates the PSF and the imaged object with blind
deconvolution. The fourth, described by Markham and Conchello
(Parametric blind deconvolution: a robust method for the
simultaneous estimation of image and blur, J. Opt. Soc. Am. 16:
2377-2391 (1999)), uses parametric blind deconvolution.
FIG. 2 shows the XZ plane of a PSF 200, wherein the acquired image
of an imaged object 240 is spread, in part, across two halves 250,
252 of an hourglass pattern. Angle 260 (.alpha.) determines the
overall shape of the hourglass pattern.
Depending upon the particular degree and direction of the spherical
aberration, a greater proportion of the acquired image may be
located in one half of the hourglass pattern than the other. For
example, as shown in FIG. 3, three generateded images are shown in
panels (a), (b), and (c), having spherical aberration coefficients
of -15, 0, and 15, respectively.
In diffraction theory, the pupil function is central to the
generation of a PSF. In the frequency domain, the optical transfer
function (OTF) is given by the pupil function convolved with its
conjugate. The OTF (in the frequency domain) and the PSF (in the
spatial domain), in turn, have a Fourier transform relationship.
Therefore, using the properties of the Fourier transform, a PSF can
be calculated as the complex multiplication of f(x,y,z) and
f*(x,y,z), where f(x,y,z) and f*(x,y,z) are the inverse Fourier
transform and its conjugate of the pupil function,
respectively.
A pupil function can be found according to the following equation
provided by Hopkins (1955), supra:
.function.eI.times..times..rho..rho..ltoreq..rho.>.times..times..times-
..times..rho..times. ##EQU00001##
k is the wave number, defined as
.times..pi..lamda. ##EQU00002##
.lamda. is the emission wavelength, typically between 350 and 1000
nm for optical sectioning;
z is the distance from the in-focus to the out-of-focus plane;
and
A is a coefficient derived from a refraction index (n) and
numerical aperture (NA).
The hourglass angle, .alpha., of the PSF is determined by the size
of the aperture (assumed to be circular) of the optical system and
the distance between the optical system and the imaged object. This
angle can be calculated according to the equation:
.times..times..alpha..times. ##EQU00003## where NA is the numerical
aperture of the optical system; and
n is the refraction index.
In theory, the hourglass angle calculated by equation 3 is the same
as the hourglass angle of the PSF calculated using equation 2.
Often, however, this is not so. Equation 3 has been used,
therefore, to validate the accuracy of the PSF calculated using
equation 2.
In addition, in order for the pupil function calculated using
equation 2 to be correct, two conditions must be met. First, X and
Y must be normalized so that the circular pupil resides exactly
within a unit circle. The normalization factor should be the
bandwidth of the pupil function, given by the equation:
.lamda..times..times. ##EQU00004## where NA is the numerical
aperture; and
.lamda. is the emission wavelength.
Second, the sampling density (.DELTA.x and .DELTA.y) in the spatial
domain must satisfy the Nyquist criterion, i.e., max (.DELTA.x,
.DELTA.y).ltoreq.(1/(2 B.sub.pupil)). Because sampling density is
determined by a user when acquiring data, the Nyquist criterion may
not be satisfied. The effect of a sampling density not satisfying
the Nyquist criterion is a PSF having a significantly smaller
hourglass angle than it should.
A further difficulty in using equation 2 to calculate a PSF is that
it assumes that PSFs are symmetric. Aberrations can cause a PSF to
be asymmetric. Wilson ("The role of the pinhole in confocal imaging
system" in HANDBOOK OF BIOLOGICAL CONFOCAL MICROSCOPY, 2d ed.
(1995)) provides the following formula for calculating a pupil
function useful in determining an asymmetric PSF: F.sub.SA(X,
Y)=F(X,Y).times.e.sup.{i2.pi..times.SA.times..rho..sup.4.sup.} (eq.
5) where
F.sub.SA(X,Y).ident. the pupil function for the spherically
aberrated PSF;
F(x,y).ident. the symmetric pupil function calculated using
Hopkins' formula above; and
SA.ident. the coefficient for spherical aberrations.
In practice, however, none of the schemes mentioned above has
proved satisfactory in correcting spherical aberration. PSFs
constructed according to diffraction theory are generally not
accurate; measured PSFs are difficult to obtain and unreliable;
PSFs estimated concurrently with an estimation of the imaged object
do not follow accurate parametric modeling of the PSF; and
parametric blind deconvolution is prohibitively slow and requires
enormous computing power.
Accordingly, a need exists for methods, systems, and program
products that quickly detect spherical aberration, provide accurate
PSF values, and utilize a robust deconvolution algorithm that
incorporates those PSF values.
SUMMARY OF THE INVENTION
The present invention provides a method, a system, and a program
product for the rapid detection and correction of spherical
aberration in microscopy systems. More specifically, the present
invention automatically detects the coefficient for spherical
aberration, empirically derives a pupil function, derives a PSF
value from the pupil function, and utilizes a robust deconvolution
algorithm that incorporates the derived PSF value.
A first aspect of the invention provides a method for detecting and
estimating spherical aberration in an image obtained using an
optical system, comprising the steps of deconvolving an acquired
image using each of a plurality of point spread functions, wherein
each point spread function has a different spherical aberration
value, calculating an image energy for each deconvolved image, and
choosing as a spherical aberration coefficient the spherical
aberration value corresponding to the deconvolved image having the
lowest image energy, wherein a spherical aberration coefficient
other than 0 indicates the presence of spherical aberration in the
acquired image and its difference from 0 is an estimation of the
degree and direction of spherical aberration.
A second aspect of the invention provides a method for detecting
and estimating spherical aberration in an image obtained using an
optical system, comprising the steps of generating a plurality of
spherically aberrated PSFs for an acquired image, wherein each PSF
has a different spherical aberration value, calculating an XZ
sum-of-intensities projection for each PSF, choosing an XZ and a YZ
cross section of the image corresponding to a maximum XZ intensity,
calculating a correlation coefficient using the PSF XZ projection
and the XZ and YZ cross-sections for each PSF, and choosing as a
spherical aberration coefficient the spherical aberration value
associated with the PSF corresponding to the highest correlation
coefficient, wherein a spherical aberration coefficient other than
0 indicates the presence of spherical aberration in the acquired
image and its difference from 0 is an estimation of the degree and
direction of spherical aberration.
A third aspect of the invention provides a method for empirically
deriving a point spread function comprising the steps of defining a
range of hourglass angles, calculating a pupil function for each
hourglass angle using the equation
.function.eI.function..times..times..rho..times..pi..rho..rho..ltoreq..rh-
o.> ##EQU00005## where k is a wave number, n is a refraction
index, z is a distance from an in-focus to an out-of-focus plane, S
is a spherical aberration value, and A is a value that will yield
the hourglass angle, fitting each value for A to a polynomial
equation, and deriving a point spread function from the pupil
function.
A fourth aspect of the invention provides a system for detecting
and correcting spherical aberration in an image obtained using an
optical system, comprising a system for detecting and estimating
spherical aberration, a system for empirically deriving a pupil
function, a system for deriving a point spread function from the
pupil function, and a system for deconvolving an imaged object from
an acquired image.
A fifth aspect of the invention provides a program product stored
on a recordable medium for detecting and correcting spherical
aberration in an image obtained using an optical system, which when
executed, comprises at least one of a program code for detecting
and estimating spherical aberration, a program code for empirically
deriving a pupil function, a program code for deriving a point
spread function from the pupil function, and a program code for
deconvolving an imaged object from an acquired image.
The foregoing and other features of the invention will be apparent
from the following more particular description of embodiments of
the invention.
BRIEF DESCRIPTION OF THE DRAWINGS
The embodiments of this invention will be described in detail, with
reference to the following figures, wherein like designations
denote like elements, and wherein:
FIG. 1 shows the differing focal points resulting from the entry of
light rays at differing distances from a lens' center.
FIG. 2 shows the XZ plane of a point spread function (PSF) having
an hourglass shape and angle .alpha..
FIG. 3 shows the XZ planes of three images with spherical
aberration coefficients of -15, 0, and 15 in panels (a), (b), and
(c), respectively.
FIG. 4 shows the XY and XZ maximum intensity projections of a
spherically aberrated image in panels (a) and (b),
respectively.
FIG. 5 shows a graph of correlation coefficient as a function of
the spherical aberration coefficient for a correlation-based
detection of spherical aberration in the image of FIG. 4.
FIG. 6 shows a deconvolved image of the image in FIG. 4 using a
correlation-based method of spherical aberration detection.
FIG. 7 shows a graph of the relationship between image energy and
spherical aberration coefficient according to an energy
minimization method of spherical aberration detection.
FIG. 8 shows a graph of the measured hourglass angles versus
expected hourglass angles using three PSF equations, including the
empirical equation of the present invention.
FIG. 9 shows a diagram of a system for the detection and correction
of spherical aberration according to the present invention.
FIG. 10 shows a computer system in which the system of FIG. 9 may
be embodied according to the present invention.
DETAILED DESCRIPTION OF THE INVENTION
A. Spherical Aberration Detection
As noted above, users of microscopy systems are often unaware of
the presence of spherical aberration during the collection of image
data and may sometimes remain unaware that the data are so
affected. Accordingly, one objective of the present invention is
the detection of spherical aberration in image data.
Two novel methods for detecting spherical aberration are provided.
The first is a correlation-based method. The second is an energy
minimization method.
1. Correlation Method
The correlation-based method is predicated on the fact that if a
spherically aberrated point spread function (PSF) is generated
correctly and approximately the same as the true PSF, then similar
structures are likely to be observed both in the spherically
aberrated image data and in the reconstructed PSF. Therefore, the
correlation-based method of the present invention first requires
the generation of a plurality of spherically aberrated PSFs, each
employing a different spherical aberration value, and the
calculation of their XZ sum-of-intensities projections. Then, XZ
and YZ cross sections of the data are chosen which correspond to
maximum intensities in the data. A correlation coefficient is then
calculated between the PSF XZ projections and the XZ and YZ
cross-sections of the data. The spherical aberration value
associated with the PSF corresponding to the highest correlation
coefficient is chosen as the spherical aberration coefficient.
The correlation coefficient can be expressed as:
.times..times..times..times..times..times..times..times..times.
##EQU00006## where u={u.sub.i|i=1,2, . . . N} are the PSF XZ
projections and v={v.sub.i|i=1,2, . . . N} are either the XZ data
cross-sections or the YZ data cross-sections.
For example, using the acquired image of FIG. 4, wherein panel (a)
shows the XY maximum intensity projection and panel (b) shows the
XZ maximum intensity projection, the correlation-based method of
spherical aberration detection was carried out. The results are
shown in the graph of FIG. 5, wherein the maximum correlation
coefficient corresponds to a spherical aberration coefficient of
approximately 4.9.
FIG. 6 shows the image resulting from the deconvolution of the
image of FIG. 5 using a Weiner filter following the
correlation-based method of spherical aberration detection. The
improvement in image quality in both the XY (panel (a)) and XZ
(panel (b)) planes is clear when compared to the uncorrected image
of FIG. 4.
While these results demonstrate that the correlation-based method
of spherical aberration detection offers significant improvements
in image quality, it was found that the correlation-based method
was not as reliable when an acquired image contained small or very
large spherical aberrations. Accordingly, a more sensitive method
for detecting spherical aberration is desirable in at least some
circumstances.
2. Energy Minimization Method
The energy minimization method of spherical aberration detection is
based upon the indirect measurement of spherical aberration using a
measurable characteristic having a correlation to spherical
aberration. Interestingly, it was found that the image energy of a
spherically aberrated PSF (whether having positive or negative
value) is higher than that of a non-spherically aberrated PSF. For
example, referring again to FIG. 3, it is easy to see that the
image intensities of the non-spherically aberrated PSF in panel (b)
are much smoother than in either the negatively spherically
aberrated PSF in panel (a) or the positively spherically aberrated
PSF in panel (c). Therefore, if the sum of intensities is fixed,
the image energy (defined as the sum of intensity squares) will be
higher for spherically aberrated PSFs than for a non-spherically
aberrated PSF.
For example, if we assume that the sum of intensities for each of
the PSFs in FIG. 3 is 10, the energy of the image in panel (b) will
be the lowest because it exhibits less variation in image intensity
than either of the images in panels (a) and (c). The sum of
intensity squares might be, for example,
(2.sup.2+2.sup.2+3.sup.2+3.sup.2=26) for the panel (b) image while
the sum of intensity squares for the panel (a) image is
(4.sup.2+4.sup.2+1.sup.2+1.sup.2=34).
Referring to FIG. 7, a graph is shown of the relationship between
image energy and spherical aberration coefficient when the image of
FIG. 4 was deconvolved using a plurality of spherically aberrated
PSFs. As can be seen, the minimum energy of the image is obtained
when the PSF has a spherical aberration value of approximately
5.05, very near the value of 4.9 obtained using the
correlation-based method.
Accordingly, the algorithm below can be used to detect the
spherical aberration of an image using the energy minimization
method. In determining the nominal value of SA, Brent's
one-dimensional search algorithm may be used. See Brent, R. P.,
ALGORITHMS FOR MINIMIZATION WITHOUT DERIVATIVES (1973). 1.
initialize spherical aberration (SA) to SA.sub.0; 2. initialize the
current minimum energy to a large number; 3. generate a PSF using
an SA; 4. deconvolve the image with the generated PSF; 5. measure
the energy of the deconvolved image; 6. if the energy in the
deconvolved image is less than the current minimum energy, then set
the nominal SA equal to SA; 7. pick a new value for SA; and 8.
repeat steps 3-7 for all SA values in the search space.
B. Empirical PSF Equation
Another objective of the present invention is the generation of
accurate PSF equations. In theory, it is possible to do so
mathematically. In practice, however, such formulas, when they can
be generated at all, tend to be very complex.
Accordingly, the present invention provides an alternative to the
mathematical derivation of PSF equations by providing a method for
empirically deriving the equations from the image data themselves.
To accomplish this, a new pupil function equation was developed
which has proven much more accurate than those described above. The
equation is expressed as:
.function.eI.function..times..times..times..times..times..times..times..t-
imes..rho..times..pi..times..times..rho..rho..ltoreq..rho.>.times.
##EQU00007## where the additional S factor defines the severity of
the spherical aberration, a value typically between -10 and 10.
To empirically derive the PSF equation, a range of hourglass angles
is first defined. For example, as shown below in Table 1, angles
were chosen between 25 and 70 degrees, in 5 degree increments.
Then, using the new pupil function equation (Eq. 7), a value for A
is determined that would generate a PSF with a measured hourglass
angle equal to each of the angles in the range. These are the
"Required A values" in Table 1. Finally, a third-order polynomial
is fitted to the data points of Table 1. That polynomial is:
A.sub.empirical=c.sub.3.alpha..sup.3+c.sub.2.alpha..sup.2+C.sub.1.alpha.+-
c.sub.0 (Eq. 8) where .alpha. is the hourglass angle calculated
using Equation 3 and c.sub.3, c.sub.2, c.sub.1, and c.sub.0 are
obtained from the polynomial fitting.
The pupil function of Equation 7, substituting the third-order
polynomial of Equation 6 for the factor A, is the empirical PSF
equation of the present invention. It is expressed as:
.function.eI.function..function..times..alpha..times..alpha..times..alpha-
..times..times..times..rho..times..pi..times..times..times..times..rho..rh-
o..ltoreq..rho.>.times. ##EQU00008##
TABLE-US-00001 TABLE 1 Angle 25 30 35 40 45 50 55 60 65 70 Required
A value 0.12 0.17 0.24 0.31 0.42 0.52 0.67 0.84 1.09 1.35
FIG. 8 shows a graph of expected hourglass angles versus measured
hourglass angles for a range of measured angles using the empirical
equation of the present invention as well as the formulae provided
by Hopkins and Born and Wolf. See Hopkins (1955), supra; Born, M.
and Wolf E., PRINCIPLES OF OPTICS, 4th ed. (1970). The Hopkins
formula is expressed as:
.times..times..alpha..times..times. ##EQU00009##
The Born and Wolf formula is expressed as:
&.times..times..times..alpha..times. ##EQU00010## which can
similarly be expressed as:
&.times..times..times..alpha..times. ##EQU00011##
As can easily be seen from FIG. 8, the empirical PSF equation of
the present invention much more accurately estimates the correct
hourglass angle than do either the Hopkins equation or the Born and
Wolf equation, each of which exhibits increasing errors as the
hourglass angle increases.
C. Adaptive Sampling Rate
As noted above, one difficulty in using any of the pupil function
equations above is that the sampling density established by a user
during the acquisition of image data may not satisfy the Nyquist
criterion, i.e., max(.DELTA.x, .DELTA.y).ltoreq.(1/(2B.sub.pupil)).
When this occurs, the circular pupil function in the frequency
domain will not be covered by the frequency range represented by
the discrete time Fourier transform. The result is a PSF function
having a significantly smaller hourglass angle than it should.
One can increase the sampling density to satisfy the Nyquist
criterion by, for example, dividing the actual sampling spacing
along each axis by an integer. If, for example, the sampling
spacing along each axis was divided by two, the frequency range of
the discrete Fourier transform will be increased by a factor of two
and is more likely to cover the pupil function. After the PSF is
generated, it is downsampled by a factor of two to conform with the
original spacing.
There are, however, two significant problems with using such a
method. First, it is unknown how large an integer the sampling
spacing must be divided by in order to ensure satisfaction of the
Nyquist criterion. Underestimating the integer will fail to solve
the problem. Overestimating the integer leads to the second
problem, namely, redundant oversampling, which unnecessarily
increases computational power and memory.
Accordingly, the present invention utilizes an adaptive sampling
rate that (1) ensures satisfaction of the Nyquist criterion, (2)
contracts the actual sampling density by a small integer in order
to avoid interpolation operations during downsampling, and (3)
minimizes computational demands.
The minimal sampling density can be calculated using the Nyquist
criterion. Therefore, an adaptive sampling rate meeting all three
of the requirements above may be found using the following
equations of the present invention:
.DELTA..times..times..DELTA..times..times..function..DELTA..times..times.-
.DELTA..times..times..times..times..DELTA..times..times..DELTA..times..tim-
es..function..DELTA..times..times..DELTA..times..times..times..times.
##EQU00012## where .DELTA.x and .DELTA.y are the actual sampling
spacing in the x and y axes and B.sub.pupil is the bandwidth of the
pupil function in Equation 4.
These two equations permit contraction of the sampling spacing by
the smallest integer necessary to guarantee satisfaction of the
Nyquist criterion while simultaneously avoiding interpolation
during downsizing and minimizing computational demands.
Referring to FIG. 9, a diagram is shown depicting a system 900 for
recovering a representation of an imaged object 962 from an
acquired image 912 according to the present invention. Image
collection system 910 gathers image data which comprise acquired
image 912. Image collection system 910 may be any system known in
the art, including, for example, a light microscope. Data
comprising acquired image 912 enters the spherical aberration
detection and correction system 970 through data input 972.
Spherical aberration detection and correction system 970 is
comprised of four subsystems 920, 930, 940, 950. Data is first
processed by a spherical aberration detection and estimation
subsystem 920. Detection and estimation may include one or both of
the correlation-based method and the energy minimization method
described herein. Next an empirical pupil function derivation
subsystem 930 empirically derives a pupil function using data from
subsystem 920. Then, a point spread function derivation subsystem
940 derives a point spread function from the pupil function derived
by subsystem 930. Finally, a deconvolution subsystem 950
deconvolves the data using the point spread function derived by
subsystem 940. Deconvolution subsystem 950 may utilize any
deconvolution algorithm known in the art, including, for example,
blind, non-blind, and inverse filter algorithms. Data processed by
subsystems 920, 930, 940, and 950 exit system 970 through data
output 974. From system 970, data may enter an output system 960
for rendering data processed by system 970 into a useful form,
shown as imaged object 962. Imaged object 962 may take any of a
number of forms, including, for example, electronic data, a
graphical display, and a projected image.
The system of FIG. 9 may be embodied, in whole or in part, in a
computer system. Referring to FIG. 10, a computer system 1080 is
shown, generally comprising a processor 1082, memory 1086, bus
1088, input/output (I/O) interfaces 1084, and external devices or
resources, depicted as image collection system 1010 and output
system 1060. Processor 1082 may comprise a single processing unit,
or be distributed across one or more processing units in one or
more locations, e.g., on a client or server. Memory 1086 may
comprise any known type of data storage and/or transmission media,
including magnetic media, optical media, random access memory
(RAM), read-only memory (ROM), a data cache, a data object, etc.
Moreover, similar to processor 1082, memory 1086 may reside at a
single physical location, comprising one or more types of data
storage, or be distributed across a plurality of physical systems
in various forms.
I/O interfaces 1084 may comprise any system for exchanging
information to and/or from an external source. External devices
and/or resources may comprise any known type of external device,
including speakers, a CRT, LED screen, hand-held device, keyboard,
mouse, voice recognition system, speech output system, printer,
monitor or display, facsimile, pager, etc. Bus 1088 provides a
communication link between each of the components in computer
system 1080 and likewise may comprise any known type of
transmission link, including electrical, optical, wireless, etc.
Although not shown, additional components, such as cache memory,
communication systems, system software, etc., may be incorporated
into computer system 1080.
Stored in memory 1086 as a program product is spherical aberration
detection and estimation system 1070, which, as described above,
processes data collected by image collection system 1010, the
processed data ultimately being provided to output system 1060.
Although not shown, computer system 1080 may include, as an
integral part or external device, a database. Such a database may
include any system (e.g., a relational database, file system, etc.)
capable of providing storage for information under the present
invention. Such information could be derived directly from computer
system 1080, or be derived, created, or stored in any other manner,
e.g., as a result of recording historical changes, etc. As such,
the database could include one or more storage devices, such as a
magnetic disk drive or an optical disk drive. In other embodiments,
the database may include data distributed across, for example, a
local area network (LAN), wide area network (WAN), or storage area
network (SAN).
It should be appreciated that the teachings of the present
invention could be offered as a business method on a subscription
basis. For example, spherical aberration detection and correction
system 1070 could be created, maintained, supported, and/or
deployed by a service provider that offers the functions described
herein for customers.
It should also be understood that the present invention can be
realized in hardware, software, a propagated signal, or any
combination thereof. Any kind of computer/server system(s)--or
other apparatus adapted for carrying out the methods described
herein--is suited. A typical combination of hardware and software
could be a general purpose computer system with a computer program
that, when loaded and executed, carries out the respective methods
described herein. Alternatively, a specific use computer,
containing specialized hardware for carrying out one or more of the
functional tasks of the invention, could be utilized. The present
invention can also be embedded in a computer program product or a
propagated signal, which comprises all the respective features
enabling the implementation of the methods described herein, and
which--when loaded in a computer system--is able to carry out these
methods. Computer program, propagated signal, software program,
program, or software, in the present context mean any expression,
in any language, code, or notation, of a set of instructions
intended to cause a system having an information processing
capability to perform a particular function either directly or
after either or both of the following: (a) conversion to another
language, code, or notation and/or (b) reproduction in a different
material form.
While this invention has been described in conjunction with the
specific embodiments outlined above, it is evident that many
alternatives, modifications and variations will be apparent to
those skilled in the art. Accordingly, the embodiments of the
invention as set forth above are intended to be illustrative, not
limiting. Various changes may be made without departing from the
spirit and scope of the invention as defined in the following
claims.
* * * * *